URBAN FARMING COORDINATED LOGISTICSAND TRANSPORTATION OF FARM PRODUCE

M.M.Sejeso∗ and M.M. Ali †

Industry Representative

N. Hoeltz1

Study group participants

J. Boshoff, F. De Villiers, A. Roux. A.D. Maphiri, M.O. Olusanya, E.M. Thulare,T. Mutavhatsindi, S.T. Sepuru, B.Scota and C. Nhangumbe

Abstract

Urban farming is important for food security and addressing socio economicchallenges. Urban framing yields small crop sizes and is spread out throughoutthe city. Farming can be done year round under controlled conditions. There isthe challenge of collecting the farm produce and getting it to the market withinsome economic constraints. In this project we model the problem as a split multi-commodity vehicle routing problem. The model allows a coordinated logistics andtransportation of farm produce. The preliminary results obtained using a heuristicmethod (local search 2-opt technique) are promising.

1 Introduction

Around the world, urban farming has significantly increased, Johannesburg is no exceptionto this phenomenon. Food gardens are springing up on the inner city roof-top. Themotivation is due to basic food security, employment creation and a multitude of socio-economic opportunities. Unlike traditional farming, urban farming yields small crop sizesand is spread throughout the city. It can be done year round as it is farmed undercontrolled conditions. Plants are grown in greenhouses to protect them from extremetemperatures, wind and pests. The produce of farms has to be collected and transportedto the market. There is a large number of small farms in contrast to the norm of one or

∗School of Computer Sicence and Applied Mathematics, University of the Witwaters-rand, Johannesburg, Private Bag 3, Wits 2050, South Africa. email: Matthews.Sejeso@ wits.ac.za†School of Computer Science and Applied Mathematics, University of the Witwaters-

rand, Johannesburg, Private Bag 3, WITS 2050, South Africa. email: Montaz.Ali@wits.ac.za1aparate South Africa (Pty) Ltd. email: Norman.Hoeltz@aparate.co

13

14 Urban farming coordinated logistics and transportation of farm produce

two large farms. These farms are often not in close proximity with each other. As suchthere is a challenge of collecting the farm produce from the farms and getting it to themarket within some economic constraints.

The challenge is mainly due to lack of coordination among the farmers. Urban farmers,spend considerable time in coordinating logistics and transporting their produce to themarket. A sizeable amount of their produce is destroyed before it reaches its destination.In business terms this translates to a significant lost of revenue. There are a few optionsthat can be taken to address this challenge.

Firstly, a coordinated system has the potential of improving the logistical operations ofthe farmers giving them time to focus on cultivation and processing of their crops. Acoordinated system that optimizes the routes between farms during collection of produceis sought after. This problem forms the basis of the well researched vehicle routingproblem [1].

Secondly, since farming can be done throughout the year, farmers are required to optimizethe planting and harvest times. This has to be coordinated over a large number of farmsproducing a range of products. Pricing of products at different times of the year mayinfluence the amount of produce. Coordination among farmers will avoid overflow of thesame product at the same time to the market. This can in turn increase the revenue offarmers. A predictive price model can be coupled with transportation of farm producefor a better coordinated logistics among farmers.

The report is organised as follows: In Section 2, we outline the problem. In Section 3, wepresent two approaches used to model the problem, a mathematical model and a heuristicmodel. Section 4 discusses the results obtained during the Study Group.

2 Problem statement

The challenge of coordinated logistics and transportation of farm produce is modelledas a classical vehicle routing problem with additional constraints. The specific vehiclerouting problem consists of a fleet of capacitated heterogeneous vehicles available to servea set of farms. Farms have known demand of collection of different produce. Differentproduce requires different vehicles, for example, strawberries require refrigeration whichis not available in all vehicles. Thus there are two sets of vehicle capabilities: refrigeratedvehicles and non-refrigerated vehicles. A refrigerated vehicle has the flexibility of carryingany set of produce. Each farm can be visited more than once and by different vehicles. Thecollection demand of each farm may be greater than the vehicle capacity. There is a singledepot. Each vehicle must start and end its tour at the depot. The objective is to find aset of vehicle routes that serve all the farms such that the sum of produce collected in eachtour does not exceed the capacity of the vehicle and increases the economic viability oftransportation of produce. For simplicity we restrict the economic viability to minimizingthe distance travelled by each vehicle.

M.M. Sejeso and M.M. Ali 15

3 Literature review

Vehicle routing problems have been studied extensively in the literature, dating backto 1959 when Dantzig and Ramser [1] introduced an algorithmic approach for petroldeliveries. Often the problem is in the context of delivering goods (having the samevalue) from a central located depot to customers, and each customer is required to bevisited exactly once. There have been lots of variations, like the split delivery vehiclerouting problem and the multi-commodity vehicle routing problem, which relate to thepresent problem.

In the split delivery vehicle routing problem the restriction that each customer has to bevisited exactly once is removed, allowing split deliveries. The problem was introducedin the literature by Dror et al. [2, 3], who motivated that allowing split deliveries cangenerate savings. Applications of the split delivery problem includes the work of Mullaserilet.al [4] who discuss the problem of managing a fleet of trucks for distributing feed in alarge livestock ranch. In [5] Archetti and Savelsbergh consider a waste collection problemwhere vehicles have a small capacity and customers have a collection demand larger thanthe capacity. Heuristic methods for solving the split delivery vehicle routing problem arepresented in [2, 3]. Local search algorithms are proposed. For example, Archetti, Hertzand Speranza [6] present a tabu search and Archetti, Savelsberg and Speranza [7] presentan optimization based heuristic.

The classical vehicle routing problem deals with goods assuming the same value. Incontrast to the vehicle routing problem the multi-commodity problem considers goodsto be of different value. Two classes of decisions are involved in this problem: routingvehicles to customers and quantifying commodities to be collected at each customer.Research done in this area includes the work of Letchford and Salazar-Gonzalez [8] andreferences therein for further related work.

4 Model formulations

In this section, we will present the model formulations that were discussed during theStudy Group. We start by establishing the baseline model based on the local searchtechnique. This gave suboptimal results. We will then present the mathematical modelformulation based on the split multi-commodity vehicle routing problem. The model isexpected to give optimal results if it can be solved efficiently.

4.1 Mathematical model

We present the mathematical model based on the split-multi commodity vehicle routingproblem. The problem can be modelled by a graph G = (F,E), with vertices F ={0, 1, 2...n} where 0 represents the depot while other vertices represent the farms, andE is the set of edges. The edge (i, j) ∈ E denotes the transversal from farm i to farm

16 Urban farming coordinated logistics and transportation of farm produce

j with the associated cost of cij. The collection demand dim of produce m is associatedwith each farm i ∈ F − {0}. There is a limited number of v vehicles available, eachwith compartmental capacity Qmk - capacity of produce m in vehicle k. Each vehiclemust start and end its route at the depot. The collection demand of the farms mustbe satisfied, and the quantity collected in each tour cannot exceed the vehicle capacityQmk. The problem can be modelled as an integer programming problem by defining thefollowing variables:

xijk =

{1 if vehicle k travels along the edge (i, j)

0 otherwise

yimk =

{1 if vehicle k collects produce m from farm i

0 otherwise

Hence the split-multi commodity vehicle problem is modelled as:

minv∑

k=1

n∑j=0

n∑i=0

cijxijk (1)

subject to :

n∑j=0

x0jk ≤ 1 ∀k;n∑

i=0

xi0k ≤ 1 ∀k (2)

n∑i=0

v∑k=1

xijk ≥ 1 ∀j (3)

n∑i=0

xijk =n∑

i=0

xjik ∀j;∀k (4)

∑i∈S

∑j∈S

xijk ≤ |S| − 1 ∀k;S ⊆ F − {0} (5)

v∑k=1

yimk = 1 ∀i;∀m (6)

yimk ≤n∑

j=0

xijk ∀i;∀m; ∀k (7)

n∑i=0

dimyimk ≤ Qmk ∀m; ∀k (8)

n∑i=0

n∑j=0

cijxijk ≤ Cmax ∀k. (9)

M.M. Sejeso and M.M. Ali 17

The objective function (1) represents the total routing cost to be minimized. Constraints(2) - (5) are classical vehicle routing constraints: constraint (2) ensures all vehicles startand end their routes at the depot, (3) imposes that each farm is visited at least once,(4) ensures the conservation of flow, while (5) eliminates the subtours. Constraints (6)- (8) concern the collection and allocation of demand among the vehicles: constraint(6)imposes that produce m at farm i must be collected by a single vehicle, (7) imposes thatvehicle k can only collect produce m from farm i if it passes through i, and (8) ensures thequantity collected by each vehicle does not exceed the capacity of the vehicle. Constraint(9) restricts each vehicle to the maximum allowed distance of Cmax.

4.2 Heuristic approach

We adopt the k-opt as a local search method to find the optimal routes. Since we havea heterogeneous fleet with v vehicles we first cluster the farms into v clusters. All farmsthat produce products that require a refrigerator are clustered together and a refrigeratedvehicle is dedicated to serve them. The remaining farms are clustered into v − 1 groupsand each group is assigned a vehicle to serve that group. A k-mean clustering algorithmis used to cluster the farms based on their proximity.

The k-mean is one of the simplest unsupervised learning algorithms that solve the clus-tering problem. The procedure follows a simple and easy way to classify a given set ofobjects into k clusters, where k is fixed a priori. The main idea is to define k centroidsfor each cluster, in this case k = v − 1. The next step is to take each object from the setand associate it to the nearest centroid. When no object is pending, the new centroid iscomputed by taking the average of each group and repeating the process. The algorithmis composed of the following steps:

1. Place k points into the space represented by the objects that are being clustered.These points represent the initial group centroids.

2. Assign each object to the group that has the closed centroid.

3. When all objects have been assigned, recalculate the positions of the k centroids.

4. Repeat steps 2 and 3 until the centroids are no longer moving. This produces aseparation of the objects into k groups.

Once the clusters are found, the 2-opt algorithm is used on each cluster to find the optimalroute. The 2-opt involves repeated breaking of two edges and reconnecting them in otherways until no positive gain 2-opt move can be made. Given the route x, a sequence atwhich the farms are visited and two farms i and j, the main operation of the 2-opt is thefollowing swapping mechanism:

1. Take x[0] to x[i] in that order, add them to the new route.

2. Take x[i + 1] to x[j] in revere the order, add them to the new route.

3. Take x[j + 1] to x[n] in that order, add them to the new route.

18 Urban farming coordinated logistics and transportation of farm produce

The above swapping mechanism is repeated for all possible 2-opt connections in a givenroute. At each step we keep the route that gave the best objective value, which is returnedas the optimal solution by the completion of the algorithm. We present the results of theabove heuristic in the next section.

5 Results

In this section, we present the results from the heuristic approach. The problem spec-ification presented by the industry representative is as follows: We have capacitated aheterogeneous fleet of three vehicles with the following capacity - there are two non-refrigerated a vehicles with capacity two ton and one ton respectively, the third vehicleis refrigerated with capacity of half a ton. The geographical locations of 50 farms werepresented in terms their longitude and latitude coordinates. The farms span the area ofJohannesburg inner-city, Johannesburg South and Soweto, see Figure 1. The location ofa single depot situated near the flea market at the Johannesburg inner-city was presentedas longitude and latitude.

We first constructed the distance matrix between the farms using the actual distancesfrom Goggle Map API. All farms with the requirement of a refrigerated vehicle whereremoved from the list, see Figure 2 and the refrigerated vehicle is dedicated to servethem. The remaining farms where clustered into two groups, see Figure 3. The clusterwith large demand is served by the vehicle with large capacity, while the other cluster wasserved by the vehicle with small capacity. The optimal routes for each of the three groupsis presented in Figure 4. The objective is to minimize the total distance - the optimalvalue using 2-opt local search is 612.42 km.

Figure 1: Location of the fifty farms.

M.M. Sejeso and M.M. Ali 19

Figure 2: Location of all farms with requirement of a refrigerated vehicle.

Figure 3: Location of farms which do not require a refrigeratedvehicle clustered into two groups depending on demand.

Figure 4: Optimal routes for the three groups.

20 Urban farming coordinated logistics and transportation of farm produce

6 Conclusions

In this work we have modelled the problem of coordinating the transportation of farmproduce as a split multi-commodity vehicle routing problem. We have presented theheuristic model to solve the problem. We first group the farms and apply the local searchalgorithm 2-opt to find the optimal route of each group. The results can be used as abenchmark for the mathematical model formulation problem. Some work still has to bedone to improve the above results. The main points to improve are:

(1) formulating and finding an effective solution of the predictive price model for plant-ing and harvesting times,

(2) find efficient solution of the mathematical model for routing planning,

(3) possibly combine the two models to have a better coordinated logistics system amongthe farms.

References

[1] G. Dantzig and J. H. Ramser. The truck dispatching problem. Management Science,6 (1959), 80–91.

[2] M. Dror and P. Trudeau. Savings by split delivery routing. Transportaion Science, 23(1990), 141–145.

[3] M. Dror and P. Trudeau. Split delivery routing. Naval research logistics, 37 (1990),383–402.

[4] P. A. Mullaseril, M. Dror, and J. Leung. Split-delivery routing in livestork feed dis-tribution. Journal of the Operational Research Society, 48 (1997), 107–116.

[5] C. Archetti and M.W.P. Savelsbergh. Vehicle routing in the 1-skip collection problem.Journal of the Operational Research Society, 55 (2004), 717–727.

[6] C. Archetti, A. Hertz and M. G. Speranza. A tabu search algorithm for the splitdelivery vehicle routing problem. Transportation Science, 40 (2006), 64–73.

[7] C. Archetti, M.W.P. Savelsbergh and M.G. Speranza. An optimizatin based heuristicfor split delivery vehicle routing problem. Transportation Science, 42 (2008), 64–73.

[8] A. N. Letchford and J. Salazar-Gonzalez. Stronger multi-commodity flow formulationsof the capacitated vehicle routing problem. European Journal of Operational Research,244 (2015), 730–738.